Conditional Statements and Reasoning

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Transcript of Conditional Statements and Reasoning

Conditional Statements and Reasoning A conditional statement, or conditional for short, is an "if-then" statement written as: "If P, then Q". Here "P" is the preceeding circumnstance, otherwise known as the hypothesis, whereas "Q" is the consequence, also known as the conclusion. For example, in the statement, "if you give me five dollars, then I'll be your best friend." The claim, "if you give me five dollars" is the hypothesis. The result, "then I'll be your best friend" is the conclusion. What is a Conditional Statement? Altering Conditional Statements The basic conditional statement can be altered using negation by denying the original hypothesis or conclusion. This relationship can be symbolized as p q A given conditional statement can be altered by negating the statement, switching the position of the hypothesis and conclusion, or both. Negation For example: "If the weather is good, then I will go swimming." conclusion, q hypothesis, p To negate this statement, we simply deny the hypothesis and/or the conclusion So, the hypothesis "the weather is good" turns into "the weather is not good" and the conclusion "I will go swimming" turns into "I will not go swimming". The negation of a statement can be expressed with: ~ Contrapositive To form the converse of a conditional statement, switch the hypothesis and conclusion. Converse For example: "If it is raining, then the grass is wet" conclusion, q hypothesis, p To find the converse of this statement, just switch the hypothesis and conclusion around. So, the orignal conditional then becomes "If the grass is wet, then it is raining." This relationship can be expressed as: q p Inverse When you negate both the hypothesis and conclusion of a conditional statement, you form the inverse. A contrapositive is formed by both negating the hypothesis and conclusion and switching their location. "If John is sweating, then it is warm outside." For example: In order to form the contrapositive of this statement, first negate the hypothesis and conclusion. "If John is not sweating, then it is not warm outside." After you negate the whole statement, swap the hypothesis for the conclusion. "If it is not warm outside, then John is not sweating." This relationship can be expressed as: ~q ~p For example: If it rains, then the game will be canceled. conclusion, q hypothesis, p In order to find the inverse, negate both the hypothesis and conclusion. So, the statement then turns into "If it does not rain, then the game will not be canceled." This relationship can be expressed as: ~p ~q *Converses aren't always trueLet's say we have this conditional: "If it is a bat, then it is a mammal."The converse of this statement would be: "If it is a mammal, then it is a bat." The converse of this statement is, obviously, false. Not all mammals are bats, but all bats are mammals. Truth Value "If the number is odd, then it is prime." Converse: False, not all odd numbers are prime, such as 9, 15, 21, and 25. Inverse: False, there is one even prime number: 2. Contrapositive: False. Not all composite numbers are even. Decide whether the statement is true or false. If false, give a counterexample as to why it’s false. The statement is false. There is one even prime number, 2, which is only divisible by one and itself. Conjecture A conjecture is, basically, an educated guess based on known information. Reasoning There are two basic types of reasoning: inductive reasoning and deductive reasoning. Inductive Reasoning Deductive Reasoning Inductive reasoning involves drawing conlusions based off of making a conclusion off of repeated observations and examples. For example: Deductive reasoning, on the other hand, is conclusions drawing a conclusion based off of a known true statement, fact, or rule. *Conjectures may or may not be true. For example: If, on every Friday for ten weeks, a school sells hotdogs, one might make the conjecture that the school sells hotdogs every Friday. However, unless the cafeteria says outright that the school will sell hotdogs on Fridays, the school for all we know could sell pizza on the eleventh week. For example: After waking up in the morning, you see water all over the ground, one might make the conjecture that it rained last night. For example: If all band students go to after-school practice on Tuesday, and Johnathan is a band student, then Johnathan goes to after-school practice on Tuesday. Indirect Proof Indirect proof is a type of proof where if a statement is assumed false and the assumption leads to an impossibility, then the statement is true. LMN has at most one right angle. For example: Let's assume LMN has more than one right angle. That is, assume that angle L and angle M are both right angles. If M and N are both right angles, then mL = mM = 90 mL + mM + mN = 180 [The sum of the measures of the angles of any triangle is 180.] 90 + 90 + mN = 180 mN = 0.This means that there is no LMN, which completely contradicts the the assumption we made.So, the assumption that L and M are both right angles must be false. Therefore, LMN has at most one right angle. Counterexample An example that proves that a conjecture false A counterexample to the statement "all prime numbers are odd numbers" is the number 2, as it is a prime number but is not an odd number. In this example, 2 is the only possible counterexample to the statement, but only a single example is needed to contradict any statement. For example: "All prime numbers are odd numbers." Credits http://www.icoachmath.com/math_dictionary/indirect_proof.html For indirect proof: For inductive/deductive reasoning: http://www.sjsu.edu/depts/itl/graphics/induc/ind-ded.html For conditional statements in general: The geometry textbook